Discrete Set-Valued Continuity and Interpolation

Abstract

Abstract. The main question of this paper is to retrieve some continuity properties on (discrete) T0-Alexandroff spaces. One possible application, which will guide us, is the construction of the so-called "tree of shapes" (intuitively, the tree of level lines). This tree, which should allow to process maxima and minima in the same way, faces quite a number of theoretical difficulties that we propose to solve using set-valued analysis in a purely discrete setting. We also propose a way to interpret any function defined on a grid as a "continuous" function thanks to an interpolation scheme. The continuity properties are essential to obtain a quasi-linear algorithm for computing the tree of shapes in any dimension, which is exposed in a companion paper [10].

A Quasi-linear Algorithm to Compute the Tree of Shapes of nD Images

“…That is, strict cut components and their complementary sets can be handled both with the same unique connectivity, c 2n . As a consequence, the operators star and Sat commute on those sets, and we can prove [17] that:…”

“…In particular, replacing the maximum operator by the median operator in Equation 1 leads to a pure self-dual definition of the tree of shapes of 2D images [17]. Furthermore the perspectives offered by that new representation might be far from being limited to the tree of shapes computation.…”

“…We can show [17] that, with those definitions, ∀u, ∀λ, [ I(u) λ ] and [ I(u) λ ] are well-composed [9]. That is, strict cut components and their complementary sets can be handled both with the same unique connectivity, c 2n .…”

“…A formal proof of our algorithm. This paper focuses on how the proposed algorithm works and gives an insight into the reasons why it works; to give a formal proof requires a large amount of materials, the first part of which can be found in [17]. About high bit-depths data.…”

“…As a consequence the tree of shapes is not only an easy access to self-dual operators such as grain filters but it has many applications, as listed in [14] (pp. [15][16][17], and some very recent works illustrate several powerful perspectives offered by that tree (see [20,21,22], and their bibliography).…”